Eradication of Time Dependency in Leaching Data

8/10/2019 Eradication of Time Dependency in Leaching Data

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Pergamon0892--6875(97)00063-0

Minerals Engineering,Vol. 10, No. 8, pp. 859--870, 1997 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved0892-6875/97 17.00+0.00

B A T C H L E A C H I N G D A T A A N A L Y S I S : E R A D I C A T I O N O FT I M E D E P E N D E N C Y P R I O R T O S T A T I S T I C A L A N A L Y S I S

N . P O W E L L a n d M . A . J O R D A N

Cam borne School o f M ines (University o f Exeter), R edruth,Cornwall, TR15 3SE, UK. E-mail: [email protected]

(Rece &e d 4 January 1997; accepted 22 April 1997)

A B S T R A C T

The evaluation of batch experimental o f data from laboratory an d pilot testwork e.g. batchleaching of metal from sulphide ore s, ma y be util ised to determine the optimum proces sconditions. Statistical analysis techniques shouM allow fo r the time-depen dent n ature o fthe data ,zs failure to recognise a nd correct fo r such t ime dependency ma y lead toincorrect interpretation o f the data. This pa pe r illustrates a corrective technique fo r time-dependent batch data. The technique is demonstrated using a typical data set rom a batchbioleach experiment. 1997 E lsevier Science Ltd

K e y w o r d sSulphide Ores, Leaching, Bioleaching

I N T R O D U C T I O N

Prelim inary batch leach ing tests such as the dissolution of copper from a low-grade (


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860 N. Powel l and M. A. Jordal l

disso lu t ion of z inc f rom the minera l su lph ide concent ra te con ta in ing 53 .3% Zn and 10 .7% Fe . A 5% w/vpulp dens i ty, in a nu t r ien t sa lt so lu tion , was prep ared in a 250m l Er len me yer f l ask and au toc laved to s te r i li sethe f l ask conten ts . Each f lask , excep t the con t ro l f l asks which were inocula ted wi th a 10%v/v so lu t ion ofthymo l in me thanol , w ere inocula ted wi th a 10%v/v inoculum of the requ is it e cu l tu re . The nu t r ien t sa l ts o l u ti o n c o n t a i n e d M g S O 4 . 7 H 2 0 ( 0 .4 g /I ), N H 4 S O4 (0.4g/I) , K 2H PO 4.3H 20 (0.1g/ l ) a nd KC I (0. l g/1) .

Eac h f lask was sam pled da i ly and the pH of the so lu t ion ad jus ted to p rede te rmine d leve l ; pH ran ges o f1 .95-2 .00 , 1 .55-1 .60 , 1 .35-1 .40 and 1 .15-1 .20 w ere used and ad jus tment was on ly a t t empted to the upperl imi t i . e . i f so lu t ion pH was be low range then no pH ad jus tment was made .

Samples were s to red a t 4C pr io r to ana lys i s by a tomic absorp t ion spec t rophotomet ry (AAS) .

E X P E R I M E N T A L R E S U L T S

Ox ida t ion o f the spha le r i t e concen t ra te wi th the mesophi l i c cu l tu res, a t var iab le s ta rt ing so lu t ion pH va lues ,resu l ted in var iab le z inc leach ing and recov ery ra tes which a re summ ar ised in Table 1 . F igures 1 and 2show z inc concent ra t ion versus l each ing t ime for each of the d i ffe ren t sta r ting so lu t ion pH va lues . The d a tase ts in F igure 1 have b een in te rpre ted us ing a se t o f French c urves whi l s t the da ta se ts in F igure 2 h avebeen f i i t ed wi th a l ine der ived us ing the mathemat ica l t echnique of l eas t squares regress ion .

TA B L E 1 S u m m a r y o f z i n c l e a c h in g a n d r e c o v er y d a t a f r o m t h e sp h a l e r it e c o n c en t r a te .

S t a r t i n gS o l u t io n p H

M a x i m u m R a te o f Z i n cS o l u b i l i s a t i o n ( m g / l / h r )

F r e n c hC u r v e

R e g r e s s i o nA n a l y s i s

C o n t r o l 0 . 8 0 . 51 .0 7 .7 5 .4

1 .2 9 .5 6 .9

Z i n cR e c o v e r y

(%)

8.02 1 . 8

2 5 . 51 .4 28 .2 18 .0 56 .71 .6 37 .1 18 .0 50 .4

Du r ing ox ida t ion of the concent ra te the so lu t ion pH in a l l f lasks was no ted to decrease i . e . so lu t ion ac id i ty

increased , w i th the resu l t tha t on te rm ina t ion of the exper im ent the so lu t ion pH in a l l f l asks was < I . 1 . Tab le1 i l lus t ra tes tha t fo r the exper imenta l condi t ions inves t iga ted a s ta r t ing so lu t ion pH va lue of 1 .4 y ie ldedsuper ior z inc recov ery a l though the maxim um ra te o f z inc leach ing was lowe r than tha t o f a s ta rt ing so lu tionpH of 1 .6 . How eve r, i t should be no ted tha t a f te r 500 hours l each ing the ra te o f z inc d isso lu t ion decrease df rom 27 .85 m gl /hr to 10 .01 mg/1 /hr resu l ting in a low er z inc recove ry to so lu t ion . Based o n these resu l t swe would sugges t tha t l each ing the concent ra te a t a s ta r t ing so lu t ion pH of 1 .4 should maximise z increco very f rom the concen t ra te . Opt imisa t ion of p rocess parameters was no t a t t empted .

S T A T ~ C A L M E T H O D O L O G Y

The s ta t i s t i ca l methodology on which the t ime-dependent na ture o f ba tch leach ing da ta can be e rad ica tedis based on Ergun ' s t es t [4 ] which assesses theequality of slopes.In the case presen ted the leach ing da tafor each tes t condi t ion was f i t ted by a regress ion l ine and the grad ien ts o f each o f these l ines com pare d tode te rmine whether the g rad ien ts cou ld be cons idered equa l .

Had the da ta se t p resen ted s imply cons i s ted of independent co lumns of f igures then to t es t fo r s ign i f ican t


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B a t c h l e a c h i n g d a t a a ~ a l y s i s 8 6 1

di ffe rences be twe,~n them the appropr ia te s ta ti s t ica l technique would be O ne-way Analys is o f Var iance(ANO VA ), which i s jus t an ex tens ion of the t - tes t for means . Ho wever, the ba tch- leaching da ta presentedhere is time-depe~'dentand thus v io la tes a fundam enta l assumpt ion of ANO VA (or of the non-parametr ica l te rna t ives). Tim e i s e ffec t ive ly a th i rd assoc ia ted variab le and i t w ould appear tha t ' cor rec t ion ' o f theoriginal data in some form is necessi tated prior to conventional s tat is t ical test ing.

16

Z i n c C o n c e n t r a t i o n ( g / l )

1 2

I 0

x

x

4

0

F

I I I I I I I I

50 150 250 350 450 550 650 750

T i m e ( H o u r s )

C t l ( 1 . 0 ) x W J ( 1 . 0 ) * W J ( 1 . 2 ) W J ( 1 . 4 ) x W J ( 1 . 6 )

Fig. 1 Zinc solubil isat ion from a sphaleri te concen trat ion using a m ixed m esoph il ic cul ture.

To eradica te the t ime-depen dency an obvious approach i s to cons ider ther a t e s of z inc ex t rac t ion and to tes tfor d i ffe rences be tween these for the vary ing so lu t ion pH va lues. The ques t ion then ar i ses as to the t imeper iod(s ) over which the ra tes for the d i ffe ren t pH va lues a re to be com pared . I f ' smal l ' t im e per iods werechosen (see F igure 3) , for example , som e form o f numer ica l d i ffe ren t ia tion could be used to es t im ate thera tes but th i s mig ht be cons idered unnecessar i ly involved .

On the o ther hand, f ile la rges t poss ib le t ime per iod which can be chosen for ca lcu la t ion of a ra te i s of coursethe range ( in t ime) o f the or ig ina l da ta it se lf . The ra te o f meta l d i sso lu tion for a par t icu lar so lu t ion pH (orthe control) then equates to the gradient or the slope of the l inear regression l ine for that pH (The l inearregression l ine, in i t. ,; ow n f ight , provides impo rtant inform ation for interpolat ion and est im ation purposes) .The appropr ia te tes would then be an 'AN OV A for s lopes ' p rovided the l inear cor re la tion coeff ic ien ts


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862 N. Pow ell and M. A. Jordan

for the d i ffe ren t pH 's were a l l deemed s ign i f ican t . Convenien t ly, a l though l i t t l e -known, such a t es t doesex is t , and i s c i ted (as an exe rc i se p rob lem) in Drap er and S mi th [5] ; the t es t appears to have be en o r ig ina l lyproposed in Ergun [4] . The m athemat ica l m ethodo logy is ou t l ined be low w hereb y the equa l i ty o f the s lopesof a se t o f regress ion l ines a re t es ted ;

16

14

12

10

Z i n c C o n c e n t r a t i o n ( g / l)

XX

X x x

X x

X

X

1

50 150 250 350 450 550 650 750

T i m e ( H o u r s )

-'-- Ct l( 1. 0) --x- WJ(1 .0) + WJ( 1.2) ~ WJ(1 .4) --x- WJ( 1.6 )

Fig .2 R egress ion l ines fo r spha le r i t e l each ing da ta .

Ergun ' s t es t assesses (as a Nul l Hypothes i s ) the equa l i ty o f the s lopes 13 of say, m l ines , represen ted byt h e m o d e l s

= - + ( 1 )

F o r i = 1 , 2 , . . m a n d j = 1 ,2 .... n .

I t is sup pose d that n i bi-varia te pairs (T i l , Y il) , (Ti2, Yi2)". .(Tini , Yin) are a vai lable for es t ima tion o f thes lopes o f the i h l ine and fur ther tha t the e r ro r o r res idua l t e rm s 8 ij fo l low aN(O,o2) distr ibut ion


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Batch leaching data analysis 863

independen tly (The m variables T i will in the specif ic case-study here al l be represented by a com mo nindependent t ime-variate T).

Zinc Concentration(g / l )7

6 -

4

0 I ' 1 I I I I I I

0 2 4 6 8 10 12 14 16 18 20

Time (Hours)

[ -- Metal Cone. (g/l) I

Fig.3 I l lustrat ion of rate determination co ncept for comparison o f rates of metal leaching betwe en data sets .This m ethod would invo lve the calculat ion o f gradients be tween al l data points on individual l ines e .g.gradients fo r the l ines AB, BC , CD etc. The method p roposed determines the gradient o f the

regression l ine f i t ted to the whole data set and uses this as a basis for comparison between data sets .

The least-squares estimate of [~ is

n i

b . - j= l ( 2 )I n l

E (rj=l

with sum of square.,; (1 degree of freedom)

n i

j l

(3)

and Res idual Sum of Squares (n -2 degrees of f reedom)


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864 N. Powell and M. A. Jordan

n i

RSS(bi) = E (Y .i-Y i)2-SS(bi) = si,s a y ( 4 ).j=

If the Null Hypothesis that~ i = 6say, for all i, is assumed true, then the least-squares estimate of I~ is

m ni

E E < ; - ~ ) < , -~ )b = i=l j=, (5 )

m ni

E E


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Batchl e a c h i n g data analysis 865

Th e Nu ll H ypothesis Ho: 13i= [3 is tested b y com paring the statistic F 2 with an app ropriate perc entage po int

F ,m-I(tabulated) of the ~ n,-2m distribution. If Ho is no t rejected then b (Eq .5) is used as the co m m on slope

of the lines.

TABLE 3 Case S tudy Da ta .

i T YI Y2 Y Y 4 Y st im e C~ l(1.0) WJ(1) W J(1.2) W J(1.4) W J(1.6)

1 66 1.659 1.499 1.059 1.611 1.5732 67 1.686 1.661 1.385 1.888 1.4723 90 1.724 2.260 1.690 2.406 1.981

4 114 1.705 2.118 1.846 2.477 2.1155 136.5 1.736 2.411 1.990 2.848 2.17 66 160.5 1.923 2.553 2.076 3.059 2.423

101112

232.75256.25280.25304.25

401424

1.9261 . 8 5 31.7791.8211.8421.770

2.8403.0443.5773.5324.0034.262

2.3922.4142.7972.8993.4633.937

4.2044.2214.7165.1218.3578.769

3.5723.7404.3194.9638.0859.1'70

13 448 1.756 3.978 4.078 8.682 8.928

14 472 1.764 3.785 3.887 8.670 9.74615 49 6 1.789 4.225 9.406 10.528.936

4.8646 592 1.921 5.061 11.177 10.80817 61 6 1.895 4.391 5.222 10.943 10.89618 66 4 2.063 5.452 5.836 13.514 13.03819 691 2.131 5.180 5.389 11.261 11.81620 736 2.124 5.575 5.289 11.767 12.41621 785 2.135 5.819 6.792 15.117 13.421

From the data

T- T ) Y: - ~ I ) = 5 5 5 . 8 8 4 ,

( T- T )( Y 3 - Z ~ ;3 ) = 7 6 2 5 . 4 4 9 ,

( T- T ) (Y 2 - ' F g 2 ) = 5 9 4 7 . 1 5 9

( T- T ) ( Y 4 - ' : '4 ) = 1 9 4 5 0 . 4 8 3

: ~ T- T ) Y s - 3 : 5 ) = 2 0 0 7 7 . 1 9 3 ,5 2 1

E E ( T- T ) ( Yi j - ~ f i ) = 5 3 6 5 6 . 1 6 8i l l j=l

Also see Table 4.

Now from the definition of sample variance,


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21

(Ti ) -T i ) 2j=l

2 0= ( 2 3 5 . 4 9 3 7 ) 2 f o r a l l i .

also sin ce T = TI = T 2 = T 3 = T 4 = T 5 giv ing

21

E (Tij -T i)j=l

= 1 1 0 9 1 4 5 . 6 5 5 f o r a l l i

and tak ing the ' c ross -product ' sum s g iven und er the case s tudy da ta ( see Table 3) and U sing Eq.2

5 5 5 . 8 8 4b I = = 0 . 0 0 0 5 0 11 1 0 9 1 4 5 . 6 5 5

5 9 4 7 . 1 5 9b 2 = = 0 . 0 0 5 3 6 2

1 1 0 9 1 4 5 . 6 5 57 6 2 5 . 4 4 9

b 3 = = 0 . 0 0 6 8 7 51 1 0 9 1 4 5 . 6 5 5

1 9 4 5 0 . 4 8 3b 4 = = 0 . 0 1 7 5 3 6

1 1 0 9 1 4 5 . 6 5 52 0 0 7 7 . 1 9 3

b~ = = 0 . 0 1 8 1 0 1" 1 1 0 9 1 4 5 . 6 5 5

Also

5 21

i=l j=l= 5 ( 1 1 0 9 1 4 5 . 6 5 5 ) = 5 5 4 5 5 7 2 8 . 2 7 5

(8)

so using Eq 5.

5 3 6 5 6 . 1 6 8b =

5 5 4 5 7 2 8 . 2 7 5= 0 . 0 0 9 6 7 5 2 2 4 8 (9)

TA BLE 4 Means an d Standard Devia t ions of Var iables.

Var i ab l e

T

Y t

Mean Sam ple Standard Devia t ion

235 .49378 2 5 01 8 5 7 2 0.14888 s t

Y2 3.6541 1 .29172Y3 3.5122 1 .63893Y4 7 .1530 4.19448Y5 7 .0089 4 .33868

23

s4S5


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Batch leaching data analysis

No w considering the calculations necessary to form the AN OV A table (see Table 2) in turn;

From Eq.6

5 21

S S ( b )= b 2E E (T/j-T/)2 =( 0 " 0 0 9 6 7 5 2 2 4 8 ) 2 ( 5 5 4 5 7 2 8 " 2 7 )i=1 j=l

8 6 7

Then

5 21

iffil j=l

5

= 2 o ( s : )i=1

= 2 0 ( 0 . 1 4 8 8 8 2 + .... 4 . 3 3 8 6 8 2) = 8 1 5 . 5 7 8 f r o m Ta b l e 4

Next (see Table 5) using Eq.3

S S ( b i )= 2 0 b i 2 ( 2 3 5 . 4 9 3 7 ) 2

TA B L E 5 Ta b l e o f S u m s o f S q u a r e Va l u e s.

i

12345

b , S $ ( b ~ )0.000501 0 .2780 .005362 31 .8890 .006875 52 .4240 .017536 52 .4240.018101 341.075

Y . 7 8 9 . 0 7 3

So from Table 2, the ANOVA table for this Case study may be drawn up as in Table 6.

TA B L E 6 A N O VA v a l u e s f o r C a s e S t u d y D a ta .

S o u r c e D e g r e e s o f S u m o f s q u ar e s M e a n S q u a re FF r e e d o m

b 1 519.135 519.135 1860.7Al l b~ g iven b 4 269.938 67.485 241.88

Res idual 95 26.505 0 .279Tota l 100 815.578

In order to test the Null hypothesis:~i = ~ for all i, the statistic F2= 241.88. Consulting table s of the F-distribution this is very highly significant (even at the 0.1% level) and therefore the Null Hypothesis must

clearly b e rejected. (The statistic F 2 = 1860.70 wo uld have bee n used to test wh ether ~ = 0 bad theHyp othesis abov e been accepted).

The next stage is to calculate the equation of the (five) regression lines, using Equation 1 with the slopesgiven by their respective estimates given in Eq.8. This is illustrated in Table 7 (means from Table 2).H I K I 0 : 1 [


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868 N. Powel l and M. A. Jordan

TA B L E

r i

0 .749 (control) 1

0.978 20.988 3

0.984 4

0.983 5

7 C a l cu l a t ed va lue s o f r eg r e s s ion l i ne s f o r Case S t u d y D a t a .

Mod el us ing Eq . 8 Re-a r range d Regre ss ion Equa t ion

Y - 1.8572 = 0.00050 I(t - 382_5) Y = 1.6655675 + 0.0 00 50 It

Y - 3.6543 = 0.005362(t - 382_5) Y = 1.6.3335 + 0.00 536 2t

Y - 3.5122 = 0.006875(t - 382_5 Y = 0.8825 125 + 0.006 875t

Y - 7.1530 = 0.017536(t - 382_5) Y = 0.44548 + 0.01 753 6t

Y - 7.0089 = 0.018 0 l(t - 382-5) Y = 0.085 2675 + 0.018 10 It

No te al l non-control v alues of r ( the correlat ion co-eff ic ient) are highly s ignif ican t even at the 0.1% level .

For purposes o f repor t ing the ca lcu la ted regress ion l ines l i st ed in Tab le 7 , and de te rmin ing the ra te o f z incsolubi l isat ion (see Table 1) , these equat ions have been quoted to 2 s ignif icant f igures (see Table 8) .

The g rad ien t and in te rcep t va lues o f each o f the equa t ions in Tab le 8 a re ' sma l l ' . Hence the num ber o fdec im al p laces quo ted and used in the ca lcu la tion o f these a re requ i red to ob ta in the requ is i te deg ree o faccuracy. I t shou ld be ev iden t tha t rounding o f in te rmedia te num er ica l va lues would ha ve resu l ted in anappreciable var ia t ion in the f inal equat ion.

TA B L E 8 F i n a l r epo r ted va lue s fo r c a l cu l a te d r eg re s s i o n equa t i ons

12

34

5

R e p o r t e d R e g r e s s io nE q u a t i o n s

Y = 1 .7 + 0 .000 50tY = 1 .6 + 0 .005 4t

Y = 0 .9 + 0 . 0 0 6 9 t

Y = 0.5-i- 0.018t

Y = 0 .09 + 0 .018 t

D I S C U S S I O N

Ergun's tes t has been appl ied to a l l the batch-leaching data with the ini t ia l nul l hypothesis that the s lopesof the (f ive) regression l ines are equal . Since one of the regression l ines wil l be f i t ted to the control data

the hop e i s obv ious ly tha t th is hypothes i s w i l l be re jec ted . I f th i s i s the case then subseque n t t e s t s cou ld becar r ied ou t fo r d i ffe rences be twee n the ra tes fo r va ry ing so lu t ion p H va lues i f requ i red . I f no t then theconc lus ion i s tha t the re i s no d i ffe rence be tween the con t ro l and tes t cond i t ions (no fu r the r ana lys i srequired) . Figure 2 i l lustra tes the regression l ines for the case s tudy data .

Us ing the p rocedure ou t l ined the re a re s t a t i s t i ca l ly s ign i f i can t d i ffe rences be tween the ra tes o f z incex t rac t ion fo r the co lum ns o f da ta (a l though a t th i s s t age i t m ay no t be poss ib le to say spec i f i ca l ly wherethe d i ffe rences l i e ). I t is ev iden t whe n exam in ing the regress ion equa t ions tha t the re i s a d i ffe rence be tweenthe control data and the rest i .e . the s ter ile acid leach and the bio-ca talysed leach ( the la t ter having a posi t iv eeffec t on m eta l l each ing ra te ) . I f th is w ere the on ly ques t ion o f in te res t a s impl i f i ca tion m igh t b e to dea lwi th jus t tw o co lum ns of da ta as be fore , one the con t ro l da ta as be fore , and the second co n ta in ing the me .ans

of the ' so lu t io n pH ' co lum ns a t the chosen t imes i. e. Ergun ' s t e s t wi th i = 2 .

Erg un ' s t e s t thus enab les p rac t ioners to dea l wi th s imi la r time-dep enden t ba tch da ta ; th i s i s obv ious ly use fu ls ince the data w il l of ten be of this type. I f it i s required to specif ical ly tes t fur the r the no n-con trol colu m nsfor d i ffe rence in the ra te o f meta l ex t rac t ion f rom one so lu tion pH, as i t is in the case p resen ted he re w herewe requ i re to iden t i fy which so lu tion pH va lue resu l t s in the ma xim um ra te o f me ta l so lub il i sa tion , then an


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Batch leachingdata analysis 869

extens ion to the procedure needs to be adopted .

The bas is of ana ly, ;i s of da ta f rom a subsequent tes t o r test s be tween regression l ines representing d i ffe ren tso lu t ion pH va lue ,, ; ma y be dependent upon fac tors ex terna l to tha t o f the da ta se t . For exam ple the co-prec ip i ta tion of z inc a t so lu t ion pH va lues in excess o f 1 .5 has been noted[6] hence even though the ra teof z inc so lubi l isa t ion may be maxim ised a t a so lu t ion pH 1 .6 opera t ing cons t ra in ts negate such condi t ions .For the case s tud y i f we assume tha t co-prec ip ita t ion of m eta ls is no t problemat ic , and the nul l hy pothes isstates that the slop,~ of the regression l ines to be tested are equal , then the next test wou ld be to d eterm inewhether or not this hypothesis holds for the control l ine and solut ion pH 1.6 l ine. If the hypothesis isrejected then this w ould indica te that there is a s ignif icant difference between the rate of zinc extract ionwhen operat ing at a solut ion pH 1.6 and the control (note that the control in each case was the samesolut ion pH under biological ly s ter i le condit ions) . A similar test could be carr ied out for each of the othersolut ion pH l ines (versus the control) to prove that there was a beneficial effect fol lowing the introductiono f a Thiobacillusculture in to the leach solut ion.

M ore impor tan t ly i s the scenar io where there v isua l ly appears to be very l i tt l e d iffe rence be tween two se tsof data e .g. the solut ion pH l ines 1.2 and 1.4 in Figure 2. In such a case Ergun's test would be applied tothese two da ta se t s a lone (exc luding the cont ro l da ta f rom which both o f these l ines had previous ly beenproven to be signit icantly different) to assess whether to accept the null hypothesis that these l ines wereequal . A resu l t ind ica t ing tha t the nul l hypothes is was t rue wo uld indica te no s igni f icant d i ffe rence be tweenthese two d a ta se t s For the two l ines ind ica ted Ergu n ' s tes t ind ica ted tha t there was no d i ffe rence .

The me thodo logy o~atl ined m ay be ta i lored to su i t ind iv idual case s tudies w here t ime-depen dent da ta requi resana lys is . In te rpre ta t ion of the resu lt s f rom E rgun ' s tes t can be based both on the da ta to be a na lysed andother fac tors both rae ta llurg ica l and econ omic .

.

.

C O N C L U S I O N S

Eradica t ion of t ime dependenc y f rom batch leaching da ta i s a requi rement pr ior to the appl ica t ionof ce rtain ~; tandard stat is tical techniqu es e.g. A NO VA .

Regress ion o f ba tch da ta fo l lowed by AN OV A on the gradien ts of these regressed l ines has beenshown/proven to d i ffe ren t ia te be tween da ta se t s for the case s tudy presented . T his technique shouldbe direct ly applicable to other instances where analysis of batch data is required.

.

2.

3.

.

5.

6.

R E F E R E N C E S

Jordan , M.A. , Pow el l , N. , Chin , C.K. & Phi l lips , C .V. , Exper imenta l Data Analys is : A Guide ToThe Select ion Of Simple Stat is t ical Tests ,Minerals Engineer ing,10, No.3, 275-286 (1997).Napier-Munn, T. , Detec t ing Per formance Improvements In Tr ia l s Wi th Time-Varying Minera lProcesses - - Three Case S tudies ,Minerals Engineer ing,8 No.8, 843-858 (1995).Jordan , M.A. , Bar r, D.W. & Phi l l ips , C .V. , I ron and Sulphur Spec ia t ion and Cel l Sur faceHyd rophob ic i ty dur ing Bacter ia l Oxida t ion of a Com plex Copper Concent ra te ,MineralsEngineer ing,6, No s 8-10 , 1001-1011 (1993).Ergu n, S. , A pplicat ion o f the principle of least squares to famil ies of s traight l ines,Industrial andEngineering Chemistry,48, 2063-2068 (Nov. 1956).Draper, N. & Smi th , H. , Appl ied Regress ion Analys is (Second Edi t ion) , John W iley & Sons Inc ,

(N ew Yo rk), ISB N 0-471-02995-5, (1981).Personal Com mun ica t ion , F.Pooley, Depar tment o f Mater ia ls Engineer ing . , Univers ity of W ales ,College of Cardiff , (1995).


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A C K N O W L E D G E M E N T S

The au thors would l ike to thank The Ins t i tu t ion o f Min ing & M eta l lu rgy fo r f inanc ia l a ss i s tance (1995Stan ley E lmore Fe l lowship Award to Dr M.A. Whi tb read-Jordan) .

Documents

Eradication of Time Dependency in Leaching Data